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Entrainment model for fully-developed wind farms: Effects of atmospheric stability and an ideal limit for wind farm performance

Paolo Luzzatto-Fegiz*

Colm-cille P. Caulfield

  • Department of Mechanical Engineering, University of California at Santa Barbara, Santa Barbara, California 93106-5070, USA

  • BP Institute for Multiphase Flow, Madingley Rise, Madingley Road, Cambridge CB3 0EZ, United Kingdom and Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom

  • *fegiz@engineering.ucsb.edu
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Phys. Rev. Fluids 3, 093802 – Published 10 September, 2018

DOI: https://doi.org/10.1103/PhysRevFluids.3.093802

Abstract

While a theoretical limit has long been established for the performance of a single turbine, no corresponding upper bound exists for the power output from a large wind farm, making it difficult to evaluate the available potential for further performance gains. Recent work involving vertical-axis turbines has achieved large increases in power density relative to traditional wind farms [J. O. Dabiri, J. Renewable Sustainable Energy 3, 043104 (2011)], thereby adding motivation to the search for an upper bound. Here, we build a model describing the essential features of a large array of turbines with arbitrary design and layout, by considering a fully-developed wind farm whose upper edge is bounded by a self-similar boundary layer. The exchanges between the wind farm, the overlaying boundary layer, and the outer flow are parametrized by means of the classical entrainment hypothesis. We obtain a concise expression for the wind farm's power density (corresponding to power output per unit planform area), as a function of three coefficients, which represent the array thrust and the turbulent exchanges at each of the two interfaces. Before seeking an upper bound on farm performance, we assess the performance of our simple model by comparing the predicted power density to field data, laboratory measurements, and large-eddy simulations for the fully-developed regions of wind farms, finding good agreement. Furthermore, we extend our model to include the effect of atmospheric stability on power output, by using a parametrization (which had been previously developed in the context of geophysical fluid dynamics) relating entrainment coefficients to local Froude numbers. Our predictions for power variation with atmospheric stability are in agreement with field measurements and large-eddy simulations. Finally, we consider an ideal limit for array operation, whereby turbines are designed to maximize momentum exchange with the overlying boundary layer. This enables us to obtain an upper bound for the performance of large wind farms, which we determine to be an order of magnitude larger than the output of contemporary turbine arrays.

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Phys. Rev. Fluids 3, 093802– Published 10 September, 2018
  • Received 19 March 2017
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